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Matrix Inversion Wiki
Definition Matrix inversion is the process of finding the matrix B''' that satisfies the prior equation for a given invertible matrix '''A. In linear algebra an n''-by-''n matrix A''' is called invertible, nonsingular, or nondegenerate, if there exists an n''-by-''n matrix '''B such that AB = BA = I''n'' where I''n'' denotes the n''-by-''n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B''' is uniquely determined by '''A and is called the inverse of A'''. Non-square matrices do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If the rank of '''A is equal to n'', then '''A' has a left inverse: BA = I'''. If '''A has rank m'', then it has a right inverse: '''AB' = I'. ''Rank of a matrix is defined as the number of linearly independent rows or columns. A square matrix is singular if and only if its determinant is 0. A Determinant is a Unique number associated with each square matrix. A general square matrix can be inverted using methods such as the'' Gaussian elimination, Gauss-Jordan elimination, or ''LU decomposition. Algorithm Gaussian elimination Gauss–Jordan elimination is an algorithm that can be used to determine whether a given matrix is invertible and to find the inverse. An alternative is the LU decomposition which generates an upper and a lower triangular matrices which are easier to invert. For special purposes, it may be convenient to invert matrices by treating mn-by-''mn'' matrices as m''-by-''m matrices of n''-by-''n matrices, and applying one or another formula recursively (other sized matrices can be padded out with dummy rows and columns). For other purposes, a variant of Newton's method may be convenient (particularly when dealing with families of related matrices, so inverses of earlier matrices can be used to seed generating inverses of later matrices). '''Example: Find the inverse of the following matrix. First, write down the entries the matrix A'', but write them in a double-wide matrix: In the other half of the double-wide, write the identity matrix: Now do matrix row operations to convert the left-hand side of the double-wide into the identity. Now that the left-hand side of the double-wide contains the identity, the right-hand side contains the inverse. That is, the inverse matrix is the following: Note that we can confirm that this matrix is the inverse of ''A by multiplying the two matrices and confirming that we get the identity: 'Inversion by using the adjoint matrix' For small matrices (2,3,4) calculating the inverse by scaling the adjoint is easier. The adjoint matrix is computed by taking the transpose of a matrix where each element is cofactor of the corresponding element in the source matrix. The cofactor is the determinant of the matrix created by taking the original matrix and removing the row and column for the element you are calculating the cofactor of. The signs of the cofactors alternate, just as when computing the determinant For example, if the original matrix M is a b c d e f g h i and the determinant is simply det(M) = a(ei-hf)-b(di - gf)+c(dh-ge). If we label the cofactors in the above array as A, B, C, etc. corresponding to the elements, the adjoint matrix would be: A D G B E H C F I The inverse of the original matrix is the adjoint, scaled by 1/det(M). Application 'Circuit' Matrix Inversion can be applied to multi-loop circuits. Applying Kirchhoff’s rules to a circuit, we obtain a set of linear equations for the currents. Consider the circuit shown in the diagram below. For junction A we have I1-I2-I3=0. For the upper loop we have V1-V2-I1R1-I2R2=0. For the lower loop we have V2-I3R3+I2R2=0. This yields the following three equations (with all quantities in SI units). A11I1+A12 I2 +A13I3=B1 with A11=1, A12=-1, A13=-1, B1=0. A21I1+A22 I2 +A23I3=B2 with A21=0.02, A22=0.05, A23=0, B2=2. A31I1+A32 I2 +A33I3=B3 with A31=0, A32=-0.05, A33=0.2, B3=10. This set of linear equations may be written a matrix equation AI=B, or or Such a matrix equation is solved by inverting the matrix, I=A-1B. 'Computer Graphics' Matrix inversion plays a significant role in computer graphics, particularly in 3D graphics rendering and 3D simulations. Examples include screen-to-world ray casting, world-to-subspace-to-world object transformations, and physical simulations. 'Other Applications' *Solving systems of n linear equations in n unknowns (a mathematical application) *Brand switching by customers (a business application) *Input-Output Models References http://electron9.phys.utk.edu/phys136d/modules/m6/m6ex1.htm - Matrix Inversion http://en.wikipedia.org/wiki/Invertible_matrix - Invertible Matrix http://people.hofstra.edu/stefan_waner/realworld/tutorialsf1/frames3_3.html - Tutorial: Matrix Inversion http://home.earthlink.net/~w6rmk/math/matinv.htm - Inverting Matrices http://www.purplemath.com/modules/mtrxinvr.htm http://mathworld.wolfram.com/MatrixInverse.html - Matrix Inverse Category:Browse